Is this vehicle a cherokee? wrangler? or a grand cherokee? What happens as i see it are two possible scenarios. One if the there is excessive resistance in that circuit the turn signals will not work, just the same way as they are for you. A poor ground, a light socket with corrosion in it will do this. Also you may have a flasher on the turn signal switch that has gone bad, this is really common. let me know which vehicle you have and I can let you know from there.
Here are the actual directions on removing, placement and install of the combination flasher. The removal numbers are messed up the second #1 is actually #3 and so on.
If this helps you get to it let me know, if you need anything else let me know. The alarm will usually malfunction indicating a problem, if you haven't had an issue with the alarm recently i wouldn't look at it right away a possible point of problem.